Joint pricing of VIX and SPX options with stochastic volatility and jump models

Abstract

Purpose – This paper studies the performance of commonly employed stochastic volatility and jump models in the consistent pricing of The CBOE Volatility Index (VIX) and The S&P 500 Index (SPX) options. With the existence of active markets for volatility derivatives and options on the underlying instrument, the need for models that are able to price these markets consistently has increased. Although pricing formulas for VIX and vanilla options are now available for commonly used models exhibiting stochastic volatility and/or jumps, it remains to be shown whether these are able to price both markets consistently. This paper fills this vacuum. Design/methodology/approach – In particular, the Heston model, the Heston model with jumps in returns and the Heston model with simultaneous jumps in returns and variance (SVJJ) are jointly calibrated to market quotes on SPX and VIX options together with VIX futures. Findings – The full flexibility of having jumps in both returns and volatility added to a stochastic volatility model is essential. Moreover, we find that the SVJJ model with the Feller condition imposed and calibrated jointly to SPX and VIX options fits both markets poorly. Relaxing the Feller condition in the calibration improves the performance considerably. Still, the fit is not satisfactory, and we conclude that one needs more flexibility in the model to jointly fit both option markets. Originality/value – Compared to existing literature, we derive numerically simpler VIX option and futures pricing formulas in the case of the SVJ model. Moreover, the paper is the first to study the pricing performance of three widely used models to SPX options and VIX derivatives.

The Quadratic Rough Heston Model and the Joint S&P 500/VIX Smile Calibration Problem

“…The last approach is to allow for jumps in the dynamic of the SPX, see [3,8,29,31,32]. Doing so, one can reconcile the skewness of SPX options with the level of VIX implied volatilities.…”

VIX-linked fees for GMWBs via explicit solution simulation methods

“…This assumption is common in the VIX derivative literature (see for example Lin (2007), Lian and Zhu (2013) and Kokholm and Stisen (2015)). In the SVJ model, the risk-neutral expectation of the log contract can be written as an affine function of the current instantaneous volatility V t :…”

“…Note that we do not differentiate betweeen the jump intensity risk and the jump size risk, and let δ * = δ. While the risk-neutral parameters of Table 1 were taken from Kokholm and Stisen (2015), the additional P -measure parameter values were chosen arbitrarily, for illustration purposes only. They yield a P -measure drift for S t that is dependent on V t , and equal to 0.0672 when V t = 0.04.…”

“…SVJ model parameters, modified fromKokholm and Stisen (2015) (calibration to SPX option data on May 26,2012) Fair fee rate.…”

Joint Modelling and Calibration of SPX and VIX by Optimal Transport

“…Other attempts include a Heston model with stochastic volatility-of-volatility by Fouque and Saporito [6] and a regimeswitching stochastic volatility model by Goutte et al [9]. In addition, many authors have tried incorporating jumps into the SPX dynamics, see, e.g., [2,4,17,19,20]. However, even with jumps, these models have yet to achieve satisfactory accuracy, particularly for short maturities.…”